5 DIRECT AND INVERSE IMAGES 7

(1.5.2) The relation A 7* 0 is equivalent to F(A) ^ 0.

(1.5.3) F({x}) = {F(%)} for every *eX.
(1 .5.4) The relation A cz B implies F(A) c F(B).

(1 .5.5) F(A n B) c F(A) n F(B).

(1 .5.6) F(A u B) = F(A) u F(B).

For F(A) c F(A u B) and F(B) cz F(A u B) by (1.5.4). On the other
hand, if y e F(A u B), there is x e A u B such that y = F(JC); as x e A or
x e B, we have y e F(A) or y e F(B).

Examples in which F(A n B) ^ F(A) n F(B) are immediate (take for
instance for F the first projection prx of a product).

For any subset A' of Y, the subset of X defined by the property ¥(x) e A'
is called the inverse image of A' by F and written F~1(A/). We have:

(1.5.7) F-1(AO = pr1(Fn(X

(1 .5.8) F-^A') = F-*(A' n F(X)), for F(JC) e F(X) is true for every x e X.

(1.5.9) P~1(0) = 0 (but here one may have F~1(A') = 0 for nonempty
subsets A', namely those for which A' n F(X) = 0).

(1 .5.10) The relation A' <= B' implies F'^A') c F^B').
(1.5.11) F'^A' n B') = F~\Af) n F'^B').

(1.5.12) F-^A' u B') = F-'CAO u F-^BO.

(1.5.13) F-1(A/-B/) = F~1(A/)-F~1(BO if A' ID B'.

Notice the difference between (1.5.11) and (1.5.5). If B c A c X, one
has by (1.5.6) F(A) = F(B) u F(A - B), hence F(A - B) => F(A) - F(B) ;
but there is no relation between F(X — A) and Y — F(A).

The set F~ *({;;}) is identical to the cross section F"1^) defined in
Section 1.3; we have:

(1.5.14) F(F~1(A/)) = A'nF(X) for A' <= Y.

(1.5.15) F-^FCA^IDA for A^X.

Finally, we note the special relations in a product:

(1.5.16) prr^A) = A x Y for any A c X; prJ^A1) = X x A' for any
A'cY.

(1 .5.1 7) C c pri(C) x pr2(C) for every C c X x Y. exists x e A such that y = F(x)" is